Optimal. Leaf size=117 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{5/2}}+\frac{x (b c-4 a d)}{3 d \sqrt{c+d x^2} (b c-a d)^2}-\frac{c x}{3 d \left (c+d x^2\right )^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.112787, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {470, 527, 12, 377, 205} \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{5/2}}+\frac{x (b c-4 a d)}{3 d \sqrt{c+d x^2} (b c-a d)^2}-\frac{c x}{3 d \left (c+d x^2\right )^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 470
Rule 527
Rule 12
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx &=-\frac{c x}{3 d (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{\int \frac{a c+(b c-3 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{3 d (b c-a d)}\\ &=-\frac{c x}{3 d (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{(b c-4 a d) x}{3 d (b c-a d)^2 \sqrt{c+d x^2}}+\frac{\int \frac{3 a^2 c d}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{3 c d (b c-a d)^2}\\ &=-\frac{c x}{3 d (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{(b c-4 a d) x}{3 d (b c-a d)^2 \sqrt{c+d x^2}}+\frac{a^2 \int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{(b c-a d)^2}\\ &=-\frac{c x}{3 d (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{(b c-4 a d) x}{3 d (b c-a d)^2 \sqrt{c+d x^2}}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{(b c-a d)^2}\\ &=-\frac{c x}{3 d (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{(b c-4 a d) x}{3 d (b c-a d)^2 \sqrt{c+d x^2}}+\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.303118, size = 160, normalized size = 1.37 \[ \frac{x^2 \left (a^2 d \left (3 c+4 d x^2\right )-a b c \left (3 c+5 d x^2\right )+b^2 c^2 x^2\right )-\frac{3 a^2 \left (c+d x^2\right )^2 \sqrt{\frac{x^2 (a d-b c)}{a c}} \tanh ^{-1}\left (\frac{\sqrt{x^2 \left (\frac{d}{c}-\frac{b}{a}\right )}}{\sqrt{\frac{d x^2}{c}+1}}\right )}{\sqrt{\frac{d x^2}{c}+1}}}{3 x \left (c+d x^2\right )^{3/2} (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 1207, normalized size = 10.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.19198, size = 1088, normalized size = 9.3 \begin{align*} \left [\frac{3 \,{\left (a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2}\right )} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} -{\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left ({\left (b c - 4 \, a d\right )} x^{3} - 3 \, a c x\right )} \sqrt{d x^{2} + c}}{12 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}, -\frac{3 \,{\left (a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2}\right )} \sqrt{\frac{a}{b c - a d}} \arctan \left (-\frac{{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt{d x^{2} + c} \sqrt{\frac{a}{b c - a d}}}{2 \,{\left (a d x^{3} + a c x\right )}}\right ) - 2 \,{\left ({\left (b c - 4 \, a d\right )} x^{3} - 3 \, a c x\right )} \sqrt{d x^{2} + c}}{6 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11974, size = 410, normalized size = 3.5 \begin{align*} -\frac{a^{2} \sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b c d - a^{2} d^{2}}} + \frac{{\left (\frac{{\left (b^{3} c^{4} d - 6 \, a b^{2} c^{3} d^{2} + 9 \, a^{2} b c^{2} d^{3} - 4 \, a^{3} c d^{4}\right )} x^{2}}{b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3} - 4 \, a^{3} b c^{2} d^{4} + a^{4} c d^{5}} - \frac{3 \,{\left (a b^{2} c^{4} d - 2 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3}\right )}}{b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3} - 4 \, a^{3} b c^{2} d^{4} + a^{4} c d^{5}}\right )} x}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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